The Statistics of Sharpe Ratios · analysis and the Sharpe–Lintner Capital Asset Pric-ing Model, the Sharpe ratio is now used in many different contexts, from performance attribution

36 ©2002, AIMR®

The Statistics of Sharpe RatiosAndrew W. Lo

The building blocks of the Sharpe ratio—expected returns and volatilities—are unknown quantities that must be estimated statistically and are,therefore, subject to estimation error. This raises the natural question: Howaccurately are Sharpe ratios measured? To address this question, I deriveexplicit expressions for the statistical distribution of the Sharpe ratio usingstandard asymptotic theory under several sets of assumptions for thereturn-generating process—independently and identically distributedreturns, stationary returns, and with time aggregation. I show thatmonthly Sharpe ratios cannot be annualized by multiplying by exceptunder very special circumstances, and I derive the correct method ofconversion in the general case of stationary returns. In an illustrativeempirical example of mutual funds and hedge funds, I find that the annualSharpe ratio for a hedge fund can be overstated by as much as 65 percentbecause of the presence of serial correlation in monthly returns, and oncethis serial correlation is properly taken into account, the rankings of hedgefunds based on Sharpe ratios can change dramatically.

ne of the most commonly cited statistics infinancial analysis is the Sharpe ratio, theratio of the excess expected return of an

investment to its return volatility or standard devi-ation. Originally motivated by mean–varianceanalysis and the Sharpe–Lintner Capital Asset Pric-ing Model, the Sharpe ratio is now used in manydifferent contexts, from performance attribution totests of market efficiency to risk management.1

Given the Sharpe ratio’s widespread use and themyriad interpretations that it has acquired over theyears, it is surprising that so little attention has beenpaid to its statistical properties. Because expectedreturns and volatilities are quantities that are gen-erally not observable, they must be estimated insome fashion. The inevitable estimation errors thatarise imply that the Sharpe ratio is also estimatedwith error, raising the natural question: How accu-rately are Sharpe ratios measured?

In this article, I provide an answer by derivingthe statistical distribution of the Sharpe ratio usingstandard econometric methods under several dif-ferent sets of assumptions for the statistical behav-ior of the return series on which the Sharpe ratio isbased. Armed with this statistical distribution, I

show that confidence intervals, standard errors,and hypothesis tests can be computed for the esti-mated Sharpe ratio in much the same way that theyare computed for regression coefficients such asportfolio alphas and betas.

The accuracy of Sharpe ratio estimators hingeson the statistical properties of returns, and theseproperties can vary considerably among portfolios,strategies, and over time. In other words, theSharpe ratio estimator’s statistical properties typi-cally will depend on the investment style of theportfolio being evaluated. At a superficial level, theintuition for this claim is obvious: The performanceof more volatile investment strategies is more dif-ficult to gauge than that of less volatile strategies.Therefore, it should come as no surprise that theresults derived in this article imply that, for exam-ple, Sharpe ratios are likely to be more accuratelyestimated for mutual funds than for hedge funds.

A less intuitive implication is that the time-series properties of investment strategies (e.g.,mean reversion, momentum, and other forms ofserial correlation) can have a nontrivial impact onthe Sharpe ratio estimator itself, especially in com-puting an annualized Sharpe ratio from monthlydata. In particular, the results derived in this articleshow that the common practice of annualizingSharpe ratios by multiplying monthly estimates by

is correct only under very special circum-stances and that the correct multiplier—whichdepends on the serial correlation of the portfolio’s

12

Andrew W. Lo is Harris & Harris Group Professor atthe Sloan School of Management, Massachusetts Insti-tute of Technology, Cambridge, and chief scientific officerfor AlphaSimplex Group, LLC, Cambridge.

O

12

The Statistics of Sharpe Ratios

July/August 2002 37

returns—can yield Sharpe ratios that are consider-ably smaller (in the case of positive serial correla-tion) or larger (in the case of negative serialcorrelation). Therefore, Sharpe ratio estimatorsmust be computed and interpreted in the context ofthe particular investment style with which a port-folio’s returns have been generated.

Let Rt denote the one-period simple return ofa portfolio or fund between dates t – 1 and t anddenote by µ and σ2 its mean and variance:

µ ≡ E(Rt), (1a)

and

σ2 ≡ Var(Rt). (1b)

Recall that the Sharpe ratio (SR) is defined as theratio of the excess expected return to the standarddeviation of return:

(2)

where the excess expected return is usually com-puted relative to the risk-free rate, Rf. Because µ andσ are the population moments of the distribution ofRt , they are unobservable and must be estimatedusing historical data.

Given a sample of historical returns (R1, R2, . . .,RT), the standard estimators for these moments arethe sample mean and variance:

(3a)

and

, (3b)

from which the estimator of the Sharpe ratio follows immediately:

. (4)

Using a set of techniques collectively known as“large-sample’’ or “asymptotic’’ statistical theoryin which the Central Limit Theorem is applied toestimators such as and , the distribution of and other nonlinear functions of and can beeasily derived.

In the next section, I present the statistical dis-tribution of under the standard assumption thatreturns are independently and identically distrib-uted (IID). This distribution completely character-izes the statistical behavior of in large samplesand allows us to quantify the precision with which

estimates SR. But because the IID assumption isextremely restrictive and often violated by financial

data, a more general distribution is derived in the“Non-IID Returns” section, one that applies toreturns with serial correlation, time-varying condi-tional volatilities, and many other characteristics ofhistorical financial time series. In the “Time Aggre-gation” section, I develop explicit expressions for“time-aggregated’’ Sharpe ratio estimators (e.g.,expressions for converting monthly Sharpe ratioestimates to annual estimates) and their distribu-tions. To illustrate the practical relevance of theseestimators, I apply them to a sample of monthlymutual fund and hedge fund returns and show thatserial correlation has dramatic effects on the annualSharpe ratios of hedge funds, inflating Sharpe ratiosby more than 65 percent in some cases and deflatingSharpe ratios in other cases.

IID ReturnsTo derive a measure of the uncertainty surroundingthe estimator , we need to specify the statisticalproperties of Rt because these properties determinethe uncertainty surrounding the component estima-tors and . Although this may seem like a theo-retical exercise best left for statisticians—not unlikethe specification of the assumptions needed to yieldwell-behaved estimates from a linear regression—there is often a direct connection between the invest-ment management process of a portfolio and itsstatistical properties. For example, a change in theportfolio manager’s style from a small-cap valueorientation to a large-cap growth orientation willtypically have an impact on the portfolio’s volatility,degree of mean reversion, and market beta. Even fora fixed investment style, a portfolio’s characteristicscan change over time because of fund inflows andoutflows, capacity constraints (e.g., a microcap fundthat is close to its market-capitalization limit),liquidity constraints (e.g., an emerging market orprivate equity fund), and changes in market condi-tions (e.g., sudden increases or decreases in volatil-ity, shifts in central banking policy, andextraordinary events, such as the default of Russiangovernment bonds in August 1998). Therefore, theinvestment style and market environment must bekept in mind when formulating the assumptions forthe statistical properties of a portfolio’s returns.

Perhaps the simplest set of assumptions that wecan specify for Rt is that they are independently andidentically distributed. This means that the proba-bility distribution of Rt is identical to that of Rs forany two dates t and s and that Rt and Rs are statisti-cally independent for all t ≠ s. Although these con-ditions are extreme and empirically implausible—the probability distribution of the monthly return ofthe S&P 500 Index in October 1987 is likely to differ

SRµ Rf–

σ-------------- ,≡

µ̂ 1T--- Rt

t =1

T

∑=

σ̂2 1T--- Rt µ̂–( )2

t =1

T

∑=

(SR)

SRµ̂ Rf–

σ̂--------------=

µ̂ σ̂2 SRµ̂ σ̂2

SR

SR

SR

SR

µ̂ σ̂2

Financial Analysts Journal

38 ©2002, AIMR®

from the probability distribution of the monthlyreturn of the S&P 500 in December 2000—they pro-vide an excellent starting point for understandingthe statistical properties of Sharpe ratios. In the nextsection, these assumptions will be replaced with amore general set of conditions for returns.

Under the assumption that returns are IID andhave finite mean µ and variance σ2, it is wellknown that the estimators and in Equation 3have the following normal distributions in largesamples, or “asymptotically,” due to the CentralLimit Theorem:2

(5)

where denotes the fact that this relationship is anasymptotic one [i.e., as T increases without bound,the probability distributions of and

approach the normal distribution, withmean zero and variances σ2 and 2σ4, respectively].These asymptotic distributions imply that the esti-mation error of and can be approximated by

(6)

where indicates that these relations are based onasymptotic approximations. Note that in Equation6, the variances of both estimators approach zero asT increases, reflecting the fact that the estimationerrors become smaller as the sample size grows. Anadditional property of and in the special caseof IID returns is that they are statistically indepen-dent in large samples, which greatly simplifies ouranalysis of the statistical properties of functions ofthese estimators (e.g., the Sharpe ratio).

Now, denote by the function g(µ,σ2) theSharpe ratio defined in Equation 2; hence, theSharpe ratio estimator is simply When the Sharpe ratio is expressed in this form, itis apparent that the estimation errors in and will affect and that the nature of theseeffects depends critically on the properties of thefunction g. Specifically, in the “IID Returns” sec-tion of Appendix A, I show that the asymptoticdistribution of the Sharpe ratio estimator is

where the asymp-totic variance is given by the following weightedaverage of the asymptotic variances of and :

(7)

The weights in Equation 7 are simply the squaredsensitivities of g with respect to µ and σ2, respec-tively: The more sensitive g is to a particular param-eter, the more influential its asymptotic variancewill be in the weighted average. This relationship

is reminiscent of the expression for the variance ofthe weighted sum of two random variables, exceptthat in Equation 7, there is no covariance term. Thisis due to the fact that and are asymptoticallyindependent, thanks to our simplifying assump-tion of IID returns. In the next sections, the IIDassumption will be replaced by a more general setof conditions on returns, in which case, the covari-ance between and will no longer be zero andthe corresponding expression for the asymptoticvariance of the Sharpe ratio estimator will be some-what more involved.

The asymptotic variance of given in Equa-tion 7 can be further simplified by evaluating thesensitivities explicitly—∂g/∂µ = 1/σ and ∂g/∂σ2 =–(µ – Rf)/(2σ3)—and then combining terms to yield

(8)

Therefore, standard errors (SEs) for the Sharpe ratioestimator can be computed as

(9)

and this quantity can be estimated by substituting for SR. Confidence intervals for SR can also be

constructed from Equation 9; for example, the 95percent confidence interval for SR around the esti-mator is simply

(10)

Table 1 reports values of Equation 9 for vari-ous combinations of Sharpe ratios and samplesizes. Observe that for any given sample size T,larger Sharpe ratios imply larger standard errors.For example, in a sample of 60 observations, thestandard error of the Sharpe ratio estimator is 0.188when the true Sharpe ratio is 1.50 but is 0.303 whenthe true Sharpe ratio is 3.00. This implies that theperformance of investments such as hedge funds,for which high Sharpe ratios are one of the primaryobjectives, will tend to be less precisely estimated.However, as a percentage of the Sharpe ratio, thestandard error given by Equation 9 does approacha finite limit as SR increases since

(11)

as SR increases without bound. Therefore, theuncertainty surrounding the IID Sharpe ratio esti-mator will be approximately the same proportion

µ̂ σ̂2

T µ̂ µ–( ) a N 0 σ2,( ) T σ̂2 σ2–( ) a N 0 2σ4,( ),, ∼∼a∼

T µ̂ µ–( )T σ̂2 σ2–( )

µ̂ σ̂2

Var µ̂( ) a σ2

T------ , Var σ̂2( ) a 2σ4

T--------- ,= =

a=

µ̂ σ̂

g µ̂ σ̂2,( ) SR.=

µ̂ σ̂2

g µ̂ σ̂2,( )

T SR – SR( ) ~a N 0 VIID,( ),

µ̂ σ̂2

VIID∂g∂µ------ 2

σ2 ∂g

∂σ2--------- 2

2σ4.+=

µ̂ σ̂2

µ̂ σ̂2

SR

VIID 1 µ Rf–( )2

2σ2----------------------+=

1 12--- SR2.+=

SR

SE(SR) =a 1 12---+ SR2

/T,

SR

SR

SR 1.96 1 12--- SR2

+ /T.×±

SE(SR)SR

------------------1 1/2( ) SR2+

T SR2---------------------------------- 1

2T-------→=


July/August 2002 39

of the Sharpe ratio for higher Sharpe ratio invest-ments with the same number of observations T.

We can develop further intuition for the impactof estimation errors in and on the Sharpe ratioby calculating the proportion of asymptotic vari-ance that is attributable to versus . From Equa-tion 7, the fraction of VIID due to estimation errorin versus is simply

(12a)

and

(12b)

For a small Sharpe ratio, such as 0.25, thisproportion—which depends only on the true Sharperatio—is 97.0 percent, indicating that most of thevariability in the Sharpe ratio estimator is a result ofvariability in . However, for higher Sharpe ratios,the reverse is true: For SR = 2.00, only 33.3 percentof the variability of comes from , and for aSharpe ratio of 3.00, only 18.2 percent of the estima-tor error of is attributable to .

Non-IID ReturnsMany studies have documented various violationsof the assumption of IID returns for financial secu-rities;3 hence, the results of the previous sectionmay be of limited practical value in certain circum-stances. Fortunately, it is possible to derive similarresults under more general conditions, conditionsthat allow for serial correlation, conditional het-eroskedasticity, and other forms of dependenceand heterogeneity in returns. In particular, if

returns satisfy the assumption of “stationarity,”then a version of the Central Limit Theorem stillapplies to most estimators and the correspondingasymptotic distribution can be derived. The formaldefinition of stationarity is that the joint probabilitydistribution of an arbitrary col-lection of returns does not changeif all the dates are incremented by the same numberof periods; that is,

(13)

for all k. Such a condition implies that mean µ andvariance σ2 (and all higher moments) are constantover time but otherwise allows for quite a broad setof dynamics for Rt, including serial correlation,dependence on such factors as the market portfolio,time-varying conditional volatilities, jumps, andother empirically relevant phenomena.

Under the assumption of stationarity,4 a ver-sion of the Central Limit Theorem can still beapplied to the estimator . However, in this case,the expression for the variance of is somewhatmore complex because of the possibility of depen-dence between the components and . In the“Non-IID Returns” section of Appendix A, I showthat the asymptotic distribution can be derived byusing a “robust’’ estimator—an estimator that iseffective under many different sets of assumptionsfor the statistical properties of returns—to estimatethe Sharpe ratio.5 In particular, I use a generalizedmethod of moments (GMM) estimator to estimate and , and the results of Hansen (1982) can be usedto obtain the following asymptotic distribution:

(14)

Table 1. Asymptotic Standard Errors of Sharpe Ratio Estimators for Combinations of Sharpe Ratio and Sample Size

Sample Size, T

SR 12 24 36 48 60 125 250 500

0.50 0.306 0.217 0.177 0.153 0.137 0.095 0.067 0.047

0.75 0.327 0.231 0.189 0.163 0.146 0.101 0.072 0.051

1.00 0.354 0.250 0.204 0.177 0.158 0.110 0.077 0.055

1.25 0.385 0.272 0.222 0.193 0.172 0.119 0.084 0.060

1.50 0.421 0.298 0.243 0.210 0.188 0.130 0.092 0.065

1.75 0.459 0.325 0.265 0.230 0.205 0.142 0.101 0.071

2.00 0.500 0.354 0.289 0.250 0.224 0.155 0.110 0.077

2.25 0.542 0.384 0.313 0.271 0.243 0.168 0.119 0.084

2.50 0.586 0.415 0.339 0.293 0.262 0.182 0.128 0.091

2.75 0.631 0.446 0.364 0.316 0.282 0.196 0.138 0.098

3.00 0.677 0.479 0.391 0.339 0.303 0.210 0.148 0.105

Note: Returns are assumed to be IID, which implies VIID = 1 + 1/2SR2.

µ̂ σ̂2

µ̂ σ̂2

µ̂ σ̂2

∂g/∂µ( )2σ2

VIID------------------------------- 1

1 1/2( ) SR2+----------------------------------=

∂g/∂σ( )22σ4

VIID----------------------------------

1/2( ) SR2

1 1/2( )SR2+---------------------------------- .=

µ̂

SR µ̂

SR µ̂

F Rt1Rt2

… Rtn, , ,( )Rt1

Rt2… Rtn

, , ,

F Rt1 + k Rt2 + k … Rtn + k, , ,( ) F Rt1Rt2

… Rtn, , ,( )=

SRSR

µ̂ σ̂2

µ̂σ̂2

T SR – SR( ) ~a N 0 VGMM,( ) VGMM,∂g∂�------ �

∂g∂�′-------- ,=


40 ©2002, AIMR®

where the definitions of ∂g/∂� and � and a methodfor estimating them are given in the second sectionof Appendix A. Therefore, for non-IID returns, thestandard error of the Sharpe ratio can be estimated by

(15)

and confidence intervals for SR can be constructedin a similar fashion to Equation 10.

Time AggregationIn many applications, it is necessary to convertSharpe ratio estimates from one frequency toanother. For example, a Sharpe ratio estimated frommonthly data cannot be directly compared with oneestimated from annual data; hence, one statisticmust be converted to the same frequency as the otherto yield a fair comparison. Moreover, in some cases,it is possible to derive a more precise estimator of anannual quantity by using monthly or daily data andthen performing time aggregation instead of esti-mating the quantity directly using annual data.6

In the case of Sharpe ratios, the most commonmethod for performing such time aggregation is tomultiply the higher-frequency Sharpe ratio by thesquare root of the number of periods contained inthe lower-frequency holding period (e.g., multiply amonthly estimator by to obtain an annual esti-mator). In this section, I show that this rule of thumbis correct only under the assumption of IID returns.For non-IID returns, an alternative procedure mustbe used, one that accounts for serial correlation inreturns in a very specific manner.

IID Returns. Consider first the case of IIDreturns. Denote by Rt(q) the following q-periodreturn:

(16)

where I have ignored the effects of compoundingfor computational convenience.7 Under the IIDassumption, the variance of Rt(q) is directly propor-tional to q; hence, the Sharpe ratio satisfies thesimple relationship:

(17)

Despite the fact that the Sharpe ratio may seemto be “unitless’’ because it is the ratio of two quan-tities with the same units, it does depend on thetimescale with respect to which the numerator anddenominator are defined. The reason is that the

numerator increases linearly with aggregationvalue q whereas the denominator increases as thesquare root of q under IID returns; hence, the ratiowill increase as the square root of q, making a longer-horizon investment seem more attractive. Thisinterpretation is highly misleading and should notbe taken at face value. Indeed, the Sharpe ratio is nota complete summary of the risks of a multiperiodinvestment strategy and should never be used as thesole criterion for making an investment decision.8

The asymptotic distribution of (q) followsdirectly from Equation 17 because (q) is propor-tional to SR:

(18)

Non-IID Returns. The relationship betweenSR and SR(q) is somewhat more involved for non-IID returns because the variance of Rt(q) is not justthe sum of the variances of component returns butalso includes all the covariances. Specifically, underthe assumption that returns Rt are stationary,

(19)

where ρk ≡ Cov(Rt, Rt–k)/Var(Rt) is the kth-orderautocorrelation of Rt.9 This yields the followingrelationship between SR and SR(q):

(20)

Note that Equation 20 reduces to Equation 17 if allautocorrelations ρk are zero, as in the case of IIDreturns. However, for non-IID returns, the adjust-ment factor for time-aggregated Sharpe ratios isgenerally not but a more complicated functionof the first q – 1 autocorrelations of returns.

Example: First-Order AutoregressiveReturns. To develop some intuition for thepotential impact of serial correlation on the Sharperatio, consider the case in which returns follow afirst-order autoregressive process or “AR(1)”:

Rt = µ + ρ(Rt–1 – µ) + εt, –1 < ρ < 1, (21)

where εt is IID with mean zero and variance σε2.

In this case, the return in period t can be forecastedto some degree by the return in period t – 1, andthis “autoregression’’ leads to serial correlation atall lags. In particular, Equation 21 implies that the

SE[SR] =a V̂GMM/T

12

Rt q( ) Rt Rt –1 … Rt – q +1,+ + +≡

SR q( )E Rt q( )[ ] Rf q( )–

Var Rt q( )[ ]-------------------------------------------=

q µ Rf–( )

qσ----------------------=

qSR.=

SRSR

T SR q( ) qSR–[ ] ~a N 0 VIID q( ), ,

VIID q( ) qVIID q 1 12--- SR2

+

.= =

Var Rt q( )[ ] Cov Rt – i Rt – j,( )j =0

q –1

∑i =0

q –1

∑=

qσ2 2σ2 q k–( )ρk,k =1

q –1

∑+=

SR q( ) η q( ) SR, η q( )q

q 2 q –1k =1

q k–( )ρk∑+------------------------------------------------------ ,≡=

q


July/August 2002 41

kth-order autocorrelation coefficient is simply ρk;hence, the scale factor in Equation 20 can be eval-uated explicitly as

(22)

Table 2 presents values of η(q) for variousvalues of ρ and q; the row corresponding to ρ = 0percent is the IID case in which the scale factor issimply . Note that for each holding-period q,positive serial correlation reduces the scale factorbelow the IID value and negative serial correlationincreases it. The reason is that positive serial corre-lation implies that the variance of multiperiodreturns increases faster than holding-period q;hence, the variance of Rt(q) is more than q times thevariance of Rt, yielding a larger denominator in theSharpe ratio than the IID case. For returns withnegative serial correlation, the opposite is true: Thevariance of Rt(q) is less than q times the variance ofRt, yielding a smaller denominator in the Sharperatio than the IID case. For returns with significantserial correlation, this effect can be substantial. Forexample, the annual Sharpe ratio of a portfolio witha monthly first-order autocorrelation of –20 percentis 4.17 times the monthly Sharpe ratio, whereas the

scale factor is 3.46 in the IID case and 2.88 when themonthly first-order autocorrelation is 20 percent.

These patterns are summarized in Figure 1, inwhich η(q) is plotted as a function of q for fivevalues of ρ. The middle (ρ = 0) curve correspondsto the standard scale factor , which is the correctη q( ) q 1

2ρ1 ρ–------------ 1

1 ρq–

q 1 ρ–( )--------------------–

+

–1/2

.=

q

Figure 1. Scale Factors of Time-Aggregated Sharpe Ratios When Returns Follow an AR(1) Process: For � = –0.50, –0.25, 0, 0.25, and 0.50

q

ρ = 0.50

ρ = 0.25

0

5

10

15

20

25

30

0 50 100 150 200 250

Aggregation Value, q

Scale Factor, η(q)

ρ = 0.50

ρ = 0.25

ρ = 0

Table 2. Scale Factors for Time-Aggregated Sharpe Ratios When Returns Follow an AR(1) Process for Various Aggregation Values and First-Order Autocorrelations

ρ(%)


2 3 4 6 12 24 36 48 125 250

90 1.03 1.05 1.07 1.10 1.21 1.41 1.60 1.77 2.67 3.70

80 1.05 1.10 1.14 1.21 1.43 1.81 2.14 2.42 3.79 5.32

70 1.08 1.15 1.21 1.33 1.65 2.19 2.62 3.00 4.75 6.68

60 1.12 1.21 1.30 1.46 1.89 2.55 3.08 3.53 5.63 7.94

50 1.15 1.28 1.39 1.60 2.12 2.91 3.53 4.06 6.49 9.15

40 1.20 1.35 1.49 1.75 2.36 3.27 3.98 4.58 7.35 10.37

30 1.24 1.43 1.60 1.91 2.61 3.65 4.44 5.12 8.23 11.62

20 1.29 1.52 1.73 2.07 2.88 4.04 4.93 5.68 9.14 12.92

10 1.35 1.62 1.86 2.25 3.16 4.45 5.44 6.28 10.12 14.31

0 1.41 1.73 2.00 2.45 3.46 4.90 6.00 6.93 11.18 15.81

–10 1.49 1.85 2.16 2.66 3.80 5.39 6.61 7.64 12.35 17.47

–20 1.58 1.99 2.33 2.90 4.17 5.95 7.31 8.45 13.67 19.35

–30 1.69 2.13 2.53 3.17 4.60 6.59 8.10 9.38 15.20 21.52

–40 1.83 2.29 2.75 3.48 5.09 7.34 9.05 10.48 17.01 24.11

–50 2.00 2.45 3.02 3.84 5.69 8.26 10.21 11.84 19.26 27.31

–60 2.24 2.61 3.37 4.30 6.44 9.44 11.70 13.59 22.19 31.50

–70 2.58 2.76 3.86 4.92 7.45 11.05 13.77 16.04 26.33 37.43

–80 3.16 2.89 4.66 5.91 8.96 13.50 16.98 19.88 32.96 47.02

–90 4.47 2.97 6.47 8.09 12.06 18.29 23.32 27.61 46.99 67.65


42 ©2002, AIMR®

factor when the correlation coefficient is zero. Thecurves above the middle one correspond to positivevalues of ρ, and those below the middle curvecorrespond to negative values of ρ. It is apparentthat serial correlation has a nontrivial effect on thetime aggregation of Sharpe ratios.

The General Case. More generally, using theexpression for (q) in Equation 20, we can con-struct an estimator of SR(q) from estimators of thefirst q – 1 autocorrelations of Rt under the assump-tion of stationary returns. As in the “Non-IIDReturn” section, we can use GMM to estimate theseautocorrelations as well as their asymptotic jointdistribution, which can then be used to derive thefollowing limiting distribution of (q):

(23)

where the definitions of ∂g/∂� and � and formulasfor estimating them are given in the “Time Aggre-gation” section of Appendix A. The standard errorof (q) is then given by

(24)

and confidence intervals can be constructed as inEquation 10.

Using When Returns Are IID.

Although the robust estimator for SR(q) is theappropriate estimator to use when returns areserially correlated or non-IID in other ways, thereis a cost: additional estimation error induced bythe autocovariance estimator, , which manifestsitself in the asymptotic variance, , of

(q). To develop a sense for the impact of esti-mation error on , consider the robustestimator when returns are, in fact, IID. In thatcase, γk = 0 for all k > 0 but because the robustestimator is a function of estimators , the estima-tion errors of the autocovariance estimators willhave an impact on . In particular, in the“Using When Returns Are IID” sectionof Appendix A, I show that for IID returns, theasymptotic variance of robust estimator (q) isgiven by

(25)

where ν3 ≡ E[(Rt – µ)3] and ν4 ≡ E[(Rt – µ)4] are thereturn’s third and fourth moments, respectively.Now suppose that returns are normally distributed.In that case, ν3 = 0 and ν4 = 3σ4, which implies that

(26)

The second term on the right side of Equation 26represents the additional estimation error intro-duced by the estimated autocovariances in themore general estimator given in Equation A18 inAppendix A. By setting q = 1 so that no time aggre-gation is involved in the Sharpe ratio estimator(hence, no autocovariances enter into the estima-tor), the expression in Equation 26 reduces to theIID case given in Equation 18.

The asymptotic relative efficiency of (q) canbe evaluated explicitly by computing the ratio ofVGMM(q) to VIID(q) in the case of IID normal returns:

(27)

and Table 3 reports these ratios for various combi-nations of Sharpe ratios and aggregation values q.Even for small aggregation values, such as q = 2,asymptotic variance VGMM(q) is significantly higherthan VIID(q)—for example, 33 percent higher for aSharpe ratio of 2.00. As the aggregation valueincreases, the asymptotic relative efficiency becomeseven worse as more estimation error is built into thetime-aggregated Sharpe ratio estimator. Even witha monthly Sharpe ratio of only 1.00, the annualized(q = 12) robust Sharpe ratio estimator has an asymp-totic variance that is 334 percent of VIID(q).

The values in Table 3 suggest that, unlessthere is significant serial correlation in returnseries Rt , the robust Sharpe ratio estimator shouldnot be used. A useful diagnostic to check for thepresence of serial correlation is the Ljung–Box(1978) Q-statistic:

(28)

which is asymptotically distributed as underthe null hypothesis of no serial correlation.10 IfQq – 1 takes on a large value—for example, if itexceeds the 95 percent critical value of the distribution—this signals significant serial corre-lation in returns and suggests that the robustSharpe ratio, (q), should be used instead of

for estimating the Sharpe ratio of q-periodreturns.

SR

SR

T SR q( ) SR q( )–[ ] ~a N 0 VGMM q( ), ,

VGMM q( )∂g∂�------ �

∂g∂�′-------- ,=

SR

SE SR q( )[ ] =a V̂GMM q( )/T

VGMM q( )

γ̂kV̂GMM q( )

SRV̂GMM q( )

γ̂k

V̂GMM q( )V̂GMM q( )

SR

VGMM q( ) 1ν3SR

σ3-------------– ν4 σ4–( )SR2

4σ4---------+=

qSR( )2 1jq--–

2

,

j =1

q –1

∑+

VGMM q( ) VIID q( ) qSR( )2 1jq--–

2

j =1

q –1

∑+=

VIID q( ).≥

SR

VGMM q( )VIID q( )

------------------------- 12 q –1

j =1 1 j/q–( )2∑1 2/SR2+

-------------------------------------------- ,+=

Qq –1 T T 2+( )ρ̂k

2

T k–------------ ,

k =1

q –1

∑=

χq –12

χq –12

SRqSR


July/August 2002 43

An Empirical ExampleTo illustrate the potential impact of estimation errorand serial correlation in computing Sharpe ratios, Iapply the estimators described in the preceding sec-tions to the monthly historical total returns of the 10largest (as of February 11, 2001) mutual funds fromvarious start dates through June 2000 and 12 hedgefunds from various inception dates through Decem-ber 2000. Monthly total returns for the mutual fundswere obtained from the University of Chicago’s Cen-ter for Research in Security Prices. The 12 hedgefunds were selected from the Altvest database toyield a diverse range of annual Sharpe ratios (from1.00 to 5.00) computed in the standard way ( ,where is the Sharpe ratio estimator applied tomonthly returns), with the additional requirementthat the funds have a minimum five-year history ofreturns. The names of the hedge funds have beenomitted to maintain their privacy, and I will refer tothem only by their investment styles (e.g., relativevalue fund, risk arbitrage fund).11

Table 4 shows that the 10 mutual funds havelittle serial correlation in returns, with p-values ofQ-statistics ranging from 13.2 percent to 80.2percent.12 Indeed, the largest absolute level ofautocorrelation among the 10 mutual funds is the12.4 percent first-order autocorrelation of theFidelity Magellan Fund. With a risk-free rate of 5/12 percent per month, the monthly Sharpe ratiosof the 10 mutual funds range from 0.14 (GrowthFund of America) to 0.32 (Janus Worldwide), withrobust standard errors of 0.05 and 0.11, respec-tively. Because of the lack of serial correlation inthe monthly returns of these mutual funds, thereis little difference between the IID estimator for the

annual Sharpe ratio, (in Table 4, ), andthe robust estimator that accounts for serialcorrelation, (12). For example, even in the caseof the Fidelity Magellan Fund, which has thehighest first-order autocorrelation among the 10mutual funds, the difference between a of0.73 and a (12) of 0.66 is not substantial (andcertainly not statistically significant). Note that therobust estimator is marginally lower than the IIDestimator, indicating the presence of positiveserial correlation in the monthly returns of theMagellan Fund. In contrast, for WashingtonMutual Investors, the IID estimate of the annualSharpe ratio is = 0.60 but the robust estimateis larger, (12) = 0.65, because of negative serialcorrelation in the fund’s monthly returns (recallthat negative serial correlation implies that thevariance of the sum of 12 monthly returns is lessthan 12 times the variance of monthly returns).

The robust standard errors SE3(12) with m = 3for (12) for the mutual funds range from 0.17(Janus) to 0.47 (Fidelity Growth and Income) andtake on similar values when m = 6, which indicatesthat the robust estimator is reasonably well behavedfor this dataset. The magnitudes of the standarderrors yield 95 percent confidence intervals forannual Sharpe ratios that do not contain 0 for anyof the 10 mutual funds. For example, the 95 percentconfidence interval for the Vanguard 500 Indexfund is 0.85 ± (1.96 × 0.26), which is (0.33, 1.36).These results indicate Sharpe ratios for the 10mutual funds that are statistically different from 0at the 95 percent confidence level.

The results for the 12 hedge funds are differentin several respects. The mean returns are higher andthe standard deviations lower, implying much

Table 3. Asymptotic Relative Efficiency of Robust Sharpe Ratio Estimator When Returns Are IID


SR 2 3 4 6 12 24 36 48 125 250

0.50 1.06 1.12 1.19 1.34 1.78 2.67 3.56 4.45 10.15 19.41

0.75 1.11 1.24 1.38 1.67 2.54 4.30 6.05 7.81 19.07 37.37

1.00 1.17 1.37 1.58 2.02 3.34 6.00 8.67 11.34 28.45 56.22

1.25 1.22 1.49 1.77 2.34 4.08 7.59 11.09 14.60 37.11 73.66

1.50 1.26 1.59 1.93 2.62 4.72 8.95 13.18 17.42 44.59 88.71

1.75 1.30 1.67 2.06 2.85 5.25 10.08 14.92 19.76 50.81 101.22

2.00 1.33 1.74 2.17 3.04 5.69 11.01 16.34 21.67 55.89 111.45

2.25 1.36 1.80 2.25 3.19 6.04 11.76 17.49 23.23 60.02 119.75

2.50 1.38 1.84 2.33 3.31 6.32 12.37 18.43 24.49 63.38 126.51

2.75 1.40 1.88 2.38 3.42 6.56 12.87 19.20 25.52 66.12 132.02

3.00 1.41 1.91 2.43 3.50 6.75 13.28 19.83 26.37 68.37 136.55

Note: Asymptotic relative efficiency is given by VGMM(q)/VIID(q).

qSRSR

qSR 12SR

SR

qSRSR

qSRSR

SR


44 ©2002, AIMR®

higher Sharpe ratio estimates for hedge funds thanfor mutual funds. The monthly Sharpe ratio esti-mates, , range from 0.56 (“Fund of funds”) to 1.26(“Convertible/option arbitrage”), in contrast to therange of 0.14 to 0.32 for the 10 mutual funds. How-ever, the serial correlation in hedge fund returns isalso much higher. For example, the first-order auto-correlation coefficient ranges from –20.2 percent to49.0 percent among the 12 hedge funds, whereas thehighest first-order autocorrelation is 12.4 percentamong the 10 mutual funds. The p-values provide amore complete summary of the presence of serial

correlation: All but 5 of the 12 hedge funds havep-values less than 5 percent, and several are less than1 percent.

The impact of serial correlation on the annualSharpe ratios of hedge funds is dramatic. When theIID estimator, , is used for the annual Sharperatio, the “Convertible/option arbitrage” fund hasa Sharpe ratio estimate of 4.35, but when serialcorrelation is properly taken into account by

(12), the estimate drops to 2.99, implying that theIID estimator overstates the annual Sharpe ratio by45 percent. The annual Sharpe ratio estimate for the

Table 4. Monthly and Annual Sharpe Ratio Estimates for a Sample of Mutual Funds and Hedge Funds

Start Date T (%) (%) (%) (%) (%)

p-Value of Q11

(%)

Monthly Annual

Fund SE3 SE3(12) SE6(12)

Mutual funds

Vanguard 500 Index 10/76 286 1.30 4.27 –4.0 –6.6 –4.9 64.5 0.21 0.06 0.72 0.85 0.26 0.25

Fidelity Magellan 1/67 402 1.73 6.23 12.4 –2.3 –0.4 28.6 0.21 0.06 0.73 0.66 0.20 0.21

Investment Company of America 1/63 450 1.17 4.01 1.8 –3.2 –4.5 80.2 0.19 0.05 0.65 0.71 0.22 0.22

Janus 3/70 364 1.52 4.75 10.5 –0.0 –3.7 58.1 0.23 0.06 0.81 0.80 0.17 0.17

Fidelity Contrafund 5/67 397 1.29 4.97 7.4 –2.5 –6.8 58.2 0.18 0.05 0.61 0.67 0.23 0.23

WashingtonMutual Investors 1/63 450 1.13 4.09 –0.1 –7.2 –2.6 22.8 0.17 0.05 0.60 0.65 0.20 0.20

Janus Worldwide 1/92 102 1.81 4.36 11.4 3.4 –3.8 13.2 0.32 0.11 1.12 1.29 0.46 0.37

Fidelity Growth and Income 1/86 174 1.54 4.13 5.1 –1.6 –8.2 60.9 0.27 0.09 0.95 1.18 0.47 0.40

American Century Ultra 12/81 223 1.72 7.11 2.3 3.4 1.4 54.5 0.18 0.07 0.64 0.71 0.27 0.25

Growth Fund of America 7/64 431 1.18 5.35 8.5 –2.7 –4.1 45.4 0.14 0.05 0.50 0.49 0.19 0.20

Hedge funds

Convertible/option arbitrage 5/92 104 1.63 0.97 42.6 29.0 21.4 0.0 1.26 0.28 4.35 2.99 1.04 1.11

Relative value 12/92 97 0.66 0.21 25.9 19.2 –2.1 4.5 1.17 0.17 4.06 3.38 1.16 1.07

Mortgage-backed securities 1/93 96 1.33 0.79 42.0 22.1 16.7 0.1 1.16 0.24 4.03 2.44 0.53 0.54

High-yield debt 6/94 79 1.30 0.87 33.7 21.8 13.1 5.2 1.02 0.27 3.54 2.25 0.74 0.72

Risk arbitrage A 7/93 90 1.06 0.69 –4.9 –10.8 6.9 30.6 0.94 0.20 3.25 3.83 0.87 0.85

Long–short equities 7/89 138 1.18 0.83 –20.2 24.6 8.7 0.1 0.92 0.06 3.19 2.32 0.35 0.37

Multistrategy A 1/95 72 1.08 0.75 48.9 23.4 3.3 0.3 0.89 0.40 3.09 2.18 1.14 1.19

Risk arbitrage B 11/94 74 0.90 0.77 –4.9 2.5 –8.3 96.1 0.63 0.14 2.17 2.47 0.79 0.77

Convertible arbitrage A 9/92 100 1.38 1.60 33.8 30.8 7.9 0.8 0.60 0.18 2.08 1.43 0.44 0.45

Convertible arbitrage B 7/94 78 0.78 0.62 32.4 9.7 –4.5 23.4 0.60 0.18 2.06 1.67 0.68 0.62

Multistrategy B 6/89 139 1.34 1.63 49.0 24.6 10.6 0.0 0.57 0.16 1.96 1.17 0.25 0.25

Fund of funds 10/94 75 1.68 2.29 29.7 21.1 0.9 23.4 0.56 0.19 1.93 1.39 0.67 0.70

Note: For the mutual fund sample, monthly total returns from various start dates through June 2000; for the hedge fund sample, variousstart dates through December 2000. The term denotes the kth autocorrelation coefficient, and Q11 denotes the Ljung–Box Q-statistic,which is asymptotically under the null hypothesis of no serial correlation. denotes the usual Sharpe ratio estimator, ,which is based on monthly data; Rf is assumed to be 5/12 percent per month; and (12) denotes the annual Sharpe ratio estimatorthat takes into account serial correlation in monthly returns. All standard errors are based on GMM estimators using the Newey–West(1982) procedure with truncation lag m = 3 for entries in the SE3 and SE3(12) columns and m = 6 for entries in the SE6(12) column.

µ̂ σ̂ ρ̂1 ρ̂2 ρ̂3SR 12SR SR(12)

ρ̂kχ11

2 SR µ̂ Rf–( )/σ̂SR

SR

12SR

SR


July/August 2002 45

“Mortgage-backed securities” fund drops from 4.03to 2.44 when serial correlation is taken into account,implying an overstatement of 65 percent. However,the annual Sharpe ratio estimate for the “Risk arbi-trage A” fund increases from 3.25 to 3.83 because ofnegative serial correlation in its monthly returns.

The sharp differences between the annual IIDand robust Sharpe ratio estimates underscore theimportance of correctly accounting for serial cor-relation in analyzing the performance of hedgefunds. Naively estimating the annual Sharpe ratiosby multiplying by will yield the rankordering given in the column of Table 4,but once serial correlation is taken into account, therank ordering changes to 3, 2, 5, 7, 1, 6, 8, 4, 10, 9,12, and 11.

The robust standard errors for the annualrobust Sharpe ratio estimates of the 12 hedge fundsrange from 0.25 to 1.16, which although larger thanthose in the mutual fund sample, neverthelessimply 95 percent confidence intervals that gener-ally do not include 0. For example, even in the caseof the “Multistrategy B” fund, which has the lowestrobust Sharpe ratio estimate (1.17), its 95 percentconfidence interval is 1.17 ± 1.96 × 0.25, which is(0.68, 1.66). These statistically significant Sharperatios are consistent with previous studies that doc-ument the fact that hedge funds do seem to exhibitstatistically significant excess returns.13 The simi-larity of the standard errors between the m = 3 andm = 6 cases for the hedge fund sample indicates thatthe robust estimator is also well behaved in thiscase, despite the presence of significant serial cor-relation in monthly returns.

In summary, the empirical examples illustratethe potential impact that serial correlation can haveon Sharpe ratio estimates and the importance ofproperly accounting for departures from the stan-dard IID framework. Robust Sharpe ratio estima-tors contain significant additional informationabout the risk–reward trade-offs for active invest-ment products, such as hedge funds; more detailedanalysis of the risks and rewards of hedge fundinvestments is performed in Getmansky, Lo, andMakarov (2002) and Lo (2001).

ConclusionAlthough the Sharpe ratio has become part of thecanon of modern financial analysis, its applicationstypically do not account for the fact that it is anestimated quantity, subject to estimation errors thatcan be substantial in some cases. The results pre-sented in this article provide one way to gauge theaccuracy of these estimators, and it should come asno surprise that the statistical properties of Sharperatios depend intimately on the statistical proper-ties of the return series on which they are based.This suggests that a more sophisticated approachto interpreting Sharpe ratios is called for, one thatincorporates information about the investmentstyle that generates the returns and the marketenvironment in which those returns are generated.For example, hedge funds have very differentreturn characteristics from the characteristics ofmutual funds; hence, the comparison of Sharperatios between these two investment vehicles can-not be performed naively. In light of the recentinterest in alternative investments by institutionalinvestors—investors that are accustomed to stan-dardized performance attribution measures suchas the annualized Sharpe ratio—there is an evengreater need to develop statistics that are consistentwith a portfolio’s investment style.

The empirical example underscores the practi-cal relevance of proper statistical inference forSharpe ratio estimators. Ignoring the impact of serialcorrelation in hedge fund returns can yield annual-ized Sharpe ratios that are overstated by more than65 percent, understated Sharpe ratios in the case ofnegatively serially correlated returns, and inconsis-tent rankings across hedge funds of different stylesand objectives. By using the appropriate statisticaldistribution for quantifying the performance of eachreturn history, the Sharpe ratio can provide a morecomplete understanding of the risks and rewards ofa broad array of investment opportunities.

I thank Nicholas Chan, Arnout Eikeboom, Jim Holub,Chris Jakob, Laurel Kenner, Frank Linet, Jon Markman,Victor Niederhoffer, Dan O’Reilly, Bill Sharpe, andJonathan Taylor for helpful comments and discussion.Research support from AlphaSimplex Group is grate-fully acknowledged.

SR 12SR 12


46 ©2002, AIMR®

Appendix A. Asymptotic Distributions of Sharpe Ratio EstimatorsThe first section of this appendix presents results for IID returns, and the second section presentscorresponding results for non-IID returns. Results for time-aggregated Sharpe ratios are reported in thethird section, and in the final section, the asymptotic variance, VGMM(q), of the time-aggregated robustestimator, (q), is derived for the special case of IID returns.

Throughout the appendix, the following conventions are maintained: (1) all vectors are column vectorsunless otherwise indicated; (2) vectors and matrixes are always typeset in boldface (i.e., X and µ are scalarsand X and � are vectors or matrixes).

IID ReturnsTo derive an expression for the asymptotic distribution of , we must first obtain the asymptotic jointdistribution of and . Denote by the column vector and let � denote the correspondingcolumn vector of population values . If returns are IID, it is a well-known consequence of the CentralLimit Theorem that the asymptotic distribution of is given by (see White):

(A1)

where the notation indicates that this is an asymptotic approximation. Because the Sharpe ratio estimator can be written as a function of , its asymptotic distribution follows directly from Taylor’s theorem

or the so-called delta method (see, for example, White):

(A2)

In the case of the Sharpe ratio, g(.) is given by Equation 2; hence,

(A3)

which yields the following asymptotic distribution for :

(A4)

Non-IID ReturnsDenote by Xt the vector of period-t returns and lags ( Rt Rt–1 . . . Rt–q+1 )′ and let (Xt) be a stochastic processthat satisfies the following conditions:

H1: {Xt : t ∈ (–∞, ∞)} is stationary and ergodic;

H2: �0 ∈ �, � is an open subset of ℜk;

H3: ∀� ∈ �, �(., �) and �θ (., �) are Borel measurable and �θ {X, .} is continuous on � for all X;

H4: �θ is first-moment continuous at �0; E[�θ(X, .)] exists, is finite, and is of full rank.

H5: Let �t ≡ �(Xt, �0)

and

vj ≡ E[�0|� – 1, � – 2, . . .] – E[�0|� – j – 1, � – j – 2, . . .]

SR

SRµ̂ σ̂2 �̂ µ̂ σ̂2( )′

µ σ2( )′�̂

T �̂ �–( ) ~a N 0 Vθ,( ) Vθ,σ2 0

0 2σ4

,≡

~a

SR g �̂( ) �̂

T g �̂( ) g �( )–[ ] ~a N 0 Vg,( ) Vg, ∂g∂�------ Vθ

∂g∂�′-------- .≡

∂g∂�′--------

1/σ

µ Rf–( )/ 2σ3( )–,=

SR

T SR – SR( ) ~a N 0 VIID,( ) VIID, 1µ Rf–( )2

2σ2----------------------+ 1 1

2--- SR2.+= =


July/August 2002 47

and assume

(i): E[�0�′0] exists and is finite,

(ii): vj converges in mean square to 0, and

(iii): (v′j vj )1/2 is finite,

which implies E[�(Xt, �0)] = 0.

H6: Let solve

Then, Hansen shows that

(A5)

where

(A6)

(A7)

and �θ(Rt, �) denotes the derivative of �(Rt, �) with respect to �.14 Specifically, let �(Rt, �) denote thefollowing vector function:

(A8)

The GMM estimator of �, denoted by , is given implicitly by the solution to

(A9)

which yields the standard estimators and given in Equation 3. For the moment conditions in EquationA8, H is given by:

. (A10)

Therefore, V� = � and the asymptotic distribution of the Sharpe ratio estimator follows from the deltamethod as in the first section:

(A11)

where ∂g/∂� is given in Equation A3. An estimator for ∂g/∂� may be obtained by substituting intoEquation A3, and an estimator for � may be obtained by using Newey and West’s (1987) procedure:

(A12)

(A13)

(A14)

E

j = 0

∞

∑

�̂1T--- � Xt �,( )

t =1

T

∑ 0.=

T �̂ �0–( ) ~a N 0 Vθ,( ) Vθ, H 1–��

1 ′–≡

H lim ET ∞→

1T--- �� Xt �0,( )

t =1

T

∑ ,≡

� lim ET ∞→

1T--- �

s =1

T

∑ Xt �0,( )� Xs �0,( )′t =1

T

∑ ,≡

� Rt �,( )Rt µ–

Rt µ–( )2 σ2–.≡

�̂

1T--- � Rt �,( )

t =1

T

∑ 0,=

µ̂ σ̂2

H lim ET ∞→

1T--- 1– 0

2 µ Rt–( ) 1–t =1

T

∑

≡ I–=

T SR – SR( ) ~a N 0 VGMM,( ) VGMM,∂g∂�------ �

∂g∂�′-------- ,=

�̂

�̂ �̂0 ω j m,( ) �̂j �′ˆj+( ), m T,«

j =1

m

∑+=

�̂j 1T--- � Rt �̂,( )� Rt – j �̂,( )′,

t = j+1

T

∑≡

ω j m,( ) 1j

m 1+-------------- ,–≡


48 ©2002, AIMR®

and m is the truncation lag, which must satisfy the condition m/T → ∞ as T increases without bound toensure consistency. An estimator for VSR can then be constructed as

(A15)

Time AggregationLet � ≡ [µ σ2 γ1 … γq–1]′ denote the vector of parameters to be estimated, where γk is the kth-orderautocovariance of Rt, and define the following moment conditions:

(A16)

where . The GMM estimator is defined by Equation A9, which yields the

standard estimators and in Equation 3 as well as the standard estimators for the autocovariances:

(A17)

The estimator for the Sharpe ratio then follows directly:

(A18)

where

As in the first two sections of this appendix, the asymptotic distribution of (q) can be obtained by applyingthe delta method to g where the function g(.) is now given by Equation 20. Recall from Equation A5 thatthe asymptotic distribution of the GMM estimator is given by

(A19)

(A20)

For the moment conditions in Equation A16, H is

; (A21)

V̂GMM∂g �( )ˆ

∂�--------------�̂

∂g �( )ˆ

∂�′-------------- .=

ϕ1 Xt �,( ) Rt µ–=

ϕ2 Xt �,( ) Rt µ–( )2 σ2–=

ϕ3 Xt �,( ) Rt µ–( ) Rt –1 µ–( ) γ1–=

ϕ4 Xt �,( ) Rt µ–( ) Rt –2 µ–( ) γ2–=

�

ϕq +1 Xt �,( ) Rt µ–( ) Rt – q +1 µ–( ) γq –1–=

� Xt �,( ) ϕ1 ϕ2 ϕ3 … ϕq +1′,≡

Xt Rt Rt –1 … Rt – q +1′≡ �̂

µ̂ σ̂2

γ̂k1T--- Rt µ̂–( ) Rt – k µ̂–( ).

t = k +1

T

∑=

SR q( ) η̂ q( ) SR, η̂ q( ) q

q 2 q –1k =1

q k–( )ρ̂k∑+------------------------------------------------------ ,≡=

ρ̂kγ̂k

σ̂2------ .=

SR�̂( )

�̂

T �̂ �–( ) ~a N 0 Vθ,( ) Vθ, H 1–�H 1– ′≡

� lim ET ∞→

1T--- � Xt �,( )� Xs �,( )′

s =1

T

∑t =1

T

∑ .≡

H lim ET ∞→

1T---

1– 0 0 … 02 µ Rt–( ) 1– 0 … 0

2µ Rt– Rt –1– 0 1– … 0

� � � �

2µ Rt– Rt – q +1– 0 … 0 1–

t =1

T

∑

I–= =...


July/August 2002 49

hence, V� = �. The asymptotic distribution of (q) then follows from the delta method:

, (A22)

where the components of ∂g/∂� are

(A23)

(A24)

(A25)

and

(A26)

Substituting into Equation A26, estimating � according to Equation A12, and forming the matrix product yields an estimator for the asymptotic variance of (q).

Using When Returns Are IID

For IID returns, it is possible to evaluate � in Equation A20 explicitly as

(A27)

where ν3 ≡ E[(Rt – µ)3], ν4 ≡ E[(Rt – µ)4], and � is partitioned into a block-diagonal matrix with a (2 × 2)matrix �1 and a diagonal (q – 1) × (q – 1) matrix �2 = σ4I along its diagonal. Because γk = 0 for all k > 0,∂g/∂� simplifies to

(A28)

(A29)

(A30)

and

(A31)

SR

T SR q( ) – SR q( )[ ] ~a N 0 VGMM q( ), VGMM q( ),∂g∂�------ �

∂g∂�′--------=

∂g∂µ------

q

σ q 2 q –1k =1

q k–( )ρk∑+----------------------------------------------------------=

∂g

∂σ2---------

q2SR

2σ2 q 2 q –1k =2

q k–( )ρk∑+3/2

-------------------------------------------------------------------------- ,–=

∂g∂γk--------

q q k–( ) SR

σ2 q 2 q –1k =1

q k–( )ρk∑+3/2

----------------------------------------------------------------------- , k =1, … q –1,,–=

∂g∂�------

∂g∂µ------

∂g∂σ2---------

∂g∂γ1-------- …

∂g∂γq –1------------- .=

�̂

∂g �̂( )/∂��∂g �̂( )/∂�′ SR

VGMM q( )

�

σ2 v3 0 0 … 0

v3 v4 σ4– 0 0 … 0

0 0 σ4 0 … �

0 0 0 σ4 … 00 0 � � �

0 0 0 0 … σ4

�1 0

0 �2

,= =

...

∂g∂µ------

qσ

------ ,=

∂g

∂σ2--------- –

qSR

2σ2-------------- ,=

∂g∂γk--------

q q k–( )SR

σ2q3/2--------------------------- , k =1, … q –1,,–=

∂g∂�------

∂g∂µ------

∂g

∂σ2----------

∂g∂γ1--------…

∂g∂γq –1-------------

a b[ ],= =


50 ©2002, AIMR®

where ∂g/∂� is also partitioned to conform to the partitioned matrix � in Equation A27. Therefore, theasymptotic variance of the robust estimator (q) is given by

(A32)

If Rt is normally distributed, then ν3 = 0 and ν4 = 3σ4; hence,

(A33)

SR

VGMM q( )∂g∂�------ �

∂g∂�′-------- a � 1a′ b�2b′+= =

q 1ν3SR

σ3-------------– ν4 σ4–( )SR2

4σ4---------+ qSR( )2 1

jq--–

2

.j =1

q –1

∑+=

VGMM q( ) q 1 3σ4 σ4–( ) SR2

4σ4---------+ qSR( )2 1

jq--–

2

j =1

q –1

∑+=

q 1 12--- SR2+

qSR( )2 1jq--–

2

j =1

q –1

∑+=

VIID q( ) qSR( )2 1jq--–

2

VIID q( ).≥j =1

q –1

∑+=


July/August 2002 51

Notes1. See Sharpe (1994) for an excellent review of its many appli-

cations, as well as some new extensions.2. The Central Limit Theorem is a remarkable mathematical

discovery on which much of modern statistical inference isbased. It shows that under certain conditions, the probabil-ity distribution of a properly normalized sum of randomvariables must converge to the standard normal distribu-tion, regardless of how each of the random variables in thesum is distributed. Therefore, using the normal distributionfor calculating significance levels and confidence intervalsis often an excellent approximation, even if normality doesnot hold for the particular random variables in question. SeeWhite (1984) for a rigorous exposition of the role of theCentral Limit Theorem in modern econometrics.

3. See, for example, Lo and MacKinlay (1999) and their cita-tions.

4. Additional regularity conditions are required; see Appen-dix A, Hansen (1982), and White for further discussion.

5. The term “robust’’ is meant to convey the ability of anestimator to perform well under various sets of assump-tions. Another commonly used term for such estimators is“nonparametric,” which indicates that an estimator is notbased on any parametric assumption, such as normallydistributed returns. See Randles and Wolfe (1979) for fur-ther discussion of nonparametric estimators and Hansenfor the generalized method of moments estimator.

6. See, for example, Campbell, Lo, and MacKinlay (1997,Ch. 9), Lo and MacKinlay (Ch. 4), Merton (1980), andShiller and Perron (1985).

7. The exact expression is, of course,

For most (but not all) applications, Equation 16 is an excel-lent approximation. Alternatively, if Rt is defined to be thecontinuously compounded return [i.e., Rt ≡ log (Pt /Pt –1),where Pt is the price or net asset value at time t], thenEquation 16 is exact.

8. See Bodie (1995) and the ensuing debate regarding risks inthe long run for further evidence of the inadequacy of theSharpe ratio—or any other single statistic—for delineatingthe risk–reward profile of a dynamic investment policy.

9. The kth-order autocorrelation of a time series Rt is definedas the correlation coefficient between Rt and Rt–k, which issimply the covariance between Rt and Rt–k divided by thesquare root of the product of the variances of Rt and Rt–k.But because the variances of Rt and Rt–k are the same underour assumption of stationarity, the denominator of theautocorrelation is simply the variance of Rt.

10. See, for example, Harvey (1981, Ch. 6.2).11. These are the investment styles reported in the Altvest

database; no attempt was made to verify or to classify thehedge funds independently.

12. The p-value of a statistic is defined as the smallest level ofsignificance for which the null hypothesis can be rejectedbased on the statistic’s value. In particular, the p-value of16.0 percent for the Q-statistic of Washington MutualInvestors in Table 4 implies that the null hypothesis of noserial correlation can be rejected only at the 16.0 percentsignificance level; at any lower level of significance—say,5 percent—the null hypothesis cannot be rejected. There-fore, smaller p-values indicate stronger evidence againstthe null hypothesis and larger p-values indicate strongerevidence in favor of the null. Researchers often reportp-values instead of test statistics because p-values are easierto interpret. To interpret a test statistic, one must compareit with the critical values of the appropriate distribution.This comparison is performed in computing the p-value.For further discussion of p-values and their interpretation,see, for example, Bickel and Doksum (1977, Ch. 5.2.B).

13. See, for example, Ackermann, McEnally, and Ravenscraft(1999), Brown, Goetzmann, and Ibbotson (1999), Brown,Goetzmann, and Park (2001), Fung and Hsieh (1997a, 1997b,2000), and Liang (1999, 2000, 2001).

14. See Magnus and Neudecker (1988) for the specific defini-tions and conventions of vector and matrix derivatives ofvector functions.

ReferencesAckermann, C., R. McEnally, and D. Ravenscraft. 1999. “ThePerformance of Hedge Funds: Risk, Return, and Incentives.”Journal of Finance, vol. 54, no. 3 (June):833–874.

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Bodie, Z. 1995. “On the Risk of Stocks in the Long Run.” FinancialAnalysts Journal, vol. 51, no. 3 (May/June):18–22.

Brown, S., W. Goetzmann, and R. Ibbotson. 1999. “OffshoreHedge Funds: Survival and Performance 1989–1995.” Journal ofBusiness, vol. 72, no. 1 (January):91–118.

Brown, S., W. Goetzmann, and J. Park. 1997. “Careers andSurvival: Competition and Risks in the Hedge Fund and CTAIndustry.” Journal of Finance, vol. 56, no. 5 (October):1869–86.

Campbell, J., A. Lo, and C. MacKinlay. 1997. The Econometrics ofFinancial Markets. Princeton, NJ: Princeton University Press.

Fung, W., and D. Hsieh. 1997a. “Empirical Characteristics ofDynamic Trading Strategies: The Case of Hedge Funds.” Reviewof Financial Studies, vol. 10, no. 2 (April):75–302.

———. 1997b. “Investment Style and Survivorship Bias in theReturns of CTAs: The Information Content of Track Records.”Journal of Portfolio Management, vol. 24, no. 1 (Fall):30–41.

———. 2000. “Performance Characteristics of Hedge Funds andCommodity Funds: Natural versus Spurious Biases.” Journal ofF inancial and Quant ita t ive Ana lys i s , vol . 35 , no. 3(September):291–307.

Getmansky, M., A. Lo, and I. Makarov. 2002. “An EconometricModel of Illiquidity and Performance Smoothing in Hedge FundReturns.” Unpublished manuscript, MIT Laboratory forFinancial Engineering.

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Rt q( ) 1 Rt –j+( ) 1.–

j =0

q –1

∏≡


52 ©2002, AIMR®

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Lo, A., and C. MacKinlay. 1999. A Non-Random Walk Down WallStreet. Princeton, NJ: Princeton University Press.

Magnus, J., and H. Neudecker. 1988. Matrix Differential Calculus:With Applications in Statistics and Economics. New York: JohnWiley & Sons.

Merton, R. 1980. “On Estimating the Expected Return on theMarket: An Exploratory Investigation.” Journal of FinancialEconomics, vol. 8, no. 4 (December):323–361.

Newey, W., and K. West. 1987. “A Simple Positive DefiniteHeteroscedasticity and Autocorrelation Consistent CovarianceMatrix.” Econometrica, vol. 55, no. 3:703–705.

Randles, R., and D. Wolfe. 1979. Introduction to the Theory ofNonparametric Statistics. New York: John Wiley & Sons.

Sharpe, W. 1994. “The Sharpe Ratio.” Journal of PortfolioManagement, vol. 21, no. 1 (Fall):49–58.

Shiller, R., and P. Perron. 1985. “Testing the Random WalkHypothesis: Power versus Frequency of Observation.”Economics Letters, vol. 18, no. 4:381–386.

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Documents

The Statistics of Sharpe Ratios · analysis and the Sharpe–Lintner Capital Asset Pric-ing Model, the Sharpe ratio is now used in many different contexts, from performance attribution